Independence

In Two Events

$P(A \bigcap B) = P(A) \times P(B)$

For example, given a coin, are the two outcomes independent?

$P( \lbrace C_1=H \rbrace \bigcap \lbrace C_2 =H \rbrace ) = 1/4$

$P( C_1=H ) \times P(C_2=H) = 1/2 \times 1/2 = 1/4$

Since the product of the two probabilities is equal to overall probability, the events are independent.

$\bigcap_i A_i = \prod_i P(A_i)$

For i $\in$ S

A & B are conditionally independent given C if the following formula holds true.

$P(A \bigcap B|C) = P(A|C) \times P(B|C)$